Functions Induced by Iterated Deformed Laguerre Derivative: Analytical and Operational Approach

Abstract

The one-parameter deformed exponential function was introduced as a frame that enchases a few known functions of this type. Such deformation requires the corresponding deformed operations (addition and subtraction) and deformed operators (derivative and antiderivative). In this paper, we will demonstrate this theory in researching of some functions defined by iterated deformed Laguerre operator. We study their properties, such as representation, orthogonality, generating function, differential and difference equation, and addition and summation formulas. Also, we consider these functions by the operational method.

1. Introduction

The several parametric generalizations and deformations of the exponential function have been proposed recently in different contexts such as nonextensive statistical mechanics [1, 2], relativistic statistical mechanics [3, 4], and quantum group theory [5–7].

The areas of deformations of the exponential functions have been treated basically along three (complementary) directions: formal mathematical developments, observation of consistent concordance with experimental (or natural) behavior, and theoretical physical developments.

In paper [8], a deformed exponential function of two variables depending on a real parameter is introduced to express discrete and continual behavior by the same. In this function, well-known generalizations and deformations can be viewed as the special cases [1, 5]. Also, its usage can be seen in [9].

In continuation of our previous considerations of the deformed exponential function, we will use its convenience to introduce and research a class of functions of two variables that can be viewed as one-parameter analog of Laguerre polynomials.

The paper is organized as follows. In Section 2, we introduce the deformed exponential function and the related deformation of variable, addition and subtraction. In Section 3, we consider some known and new difference and differential operators, convenient for the work with deformed variables and exponentials. In this environment, in Section 4, we define a class of two-variable functions, called the deformed Laguerre polynomials, by an iterated generalized differential operator. Section 5 is devoted to various properties of these functions, such as orthogonality, summation and addition formulas, differential equations, generating function. Finally, in Section 6 we prove a few operational identities involving the introduced deformed Laguerre polynomials.

Because of the presence of “logarithmic scale,” deformed Laguerre operator and deformed Laguerre polynomials could be suitable for use in control engineering, population dynamics, mathematical modeling of viscous fluids, and oscillating problems in mechanics, like the usual Laguerre operator and Laguerre polynomials that are already used (see [10–12]).

2. The Deformed Exponential Functions

In this section we will present a deformation of an exponential function of two variables depending on parameter ℎ∈ℝ⧵{0}, which is introduced in [8].

Let us define function (𝑥,𝑦)↦𝑒_ℎ(𝑥,𝑦) by 𝑒_ℎ𝑥,𝑦=(1+ℎ𝑥)^𝑦/ℎ−1𝑥∈ℂ⧵ℎ,𝑦∈ℝ. (2.1)

Since lim_ℎ→0𝑒_ℎ𝑥,𝑦=𝑒^𝑥𝑦, (2.2) this function can be viewed as a one-parameter deformation of the exponential function of two variables.

If ℎ=1−𝑞(𝑞≠1) and 𝑦=1, function (2.1) becomes 𝑒_1−𝑞(𝑥,1)=1+1−𝑞𝑥^1/(1−𝑞), (2.3) that is, 𝑒_1−𝑞(𝑥,1)=𝑒^𝑥_𝑞, where 𝑒^𝑥_𝑞 is Tsallis 𝑞-exponential function [1] defined by𝑒^𝑥_𝑞=1+1−𝑞𝑥^1/(1−𝑞)(),1+1−𝑞𝑥>00,otherwise𝑥∈ℝ. (2.4)

If ℎ=𝑝−1(𝑝≠1) and 𝑥=1, function (2.1) becomes 𝑒_𝑝−11,𝑦=𝑝^{𝑦/(𝑝−1)}, (2.5) that is, a function considered for a generalization of the standard exponential function in the context of quantum group formalism [13].

Notice that function (2.1) can be written in the form 𝑒_ℎ𝑦𝑥,𝑦=expℎln(1+ℎ𝑥). (2.6) Hence, similar to what in [14], we can use cylinder transformation as deformation function 𝑥↦{𝑥}_ℎ by {𝑥}_ℎ=1ℎln(1+ℎ𝑥)=ln(1+ℎ𝑥)^1/ℎ−1𝑥∈ℂ⧵ℎ. (2.7) Thus, the following holds: 𝑒_ℎ𝑥,𝑦=𝑒^{{𝑥}ℎ𝑦}. (2.8)

We can show that function (2.1) holds on some basic properties of the exponential function.

Proposition 2.1. For 𝑥∈ℂ⧵{−1/ℎ} and 𝑦,𝑦₁,𝑦₂∈ℝ, the following holds: 𝑒_ℎ1𝑥,𝑦>0𝑥<−ℎ1forℎ<0or𝑥>−ℎ,𝑒forℎ>0_ℎ0,𝑦=𝑒_ℎ()𝑒𝑥,0=1,_−ℎ𝑥,𝑦=𝑒_ℎ1−𝑥,−𝑦𝑥≠ℎ,𝑒_ℎ𝑥,𝑦₁+𝑦₂=𝑒_ℎ𝑥,𝑦₁𝑒_ℎ𝑥,𝑦₂. (2.9)

Notice that the additional property is true with respect to the second variable only. However, with respect to the first variable, the following holds:𝑒_ℎ𝑥₁,𝑦𝑒_ℎ𝑥₂,𝑦=𝑒_ℎ𝑥₁+𝑥₂+ℎ𝑥₁𝑥₂,𝑦. (2.10)

This equality suggests introducing a generalization of the sum operation 𝑥₁⊕_ℎ𝑥₂=𝑥₁+𝑥₂+ℎ𝑥₁𝑥₂. (2.11) Such generalized addition operator was considered in some papers and books (see, e.g., [2] or [14]). This operation is commutative and associative, and zero is its neutral. For 𝑥≠−1/ℎ, the ⊖_ℎ-inverse exists as ⊖_ℎ𝑥=−𝑥1+ℎ𝑥, (2.12) and 𝑥⊕_ℎ(⊖_ℎ𝑥)=0 is valid. Hence, (𝐼,⊕_ℎ) is an abelian group, where 𝐼=(−∞,−1/ℎ) for ℎ<0 or 𝐼=(−1/ℎ,+∞) for ℎ>0. In this way, the ⊖_ℎ-subtraction can be defined by 𝑥₁⊖_ℎ𝑥₂=𝑥₁⊕_ℎ⊖_ℎ𝑥₂=𝑥₁−𝑥₂1+ℎ𝑥₂𝑥₂1≠−ℎ. (2.13)

With respect to (2.7), we can prove the next equality for 𝑥₁,𝑥₂∈𝐼: 𝑥_1ℎ+𝑥_2ℎ=𝑥₁⊕_ℎ𝑥_2ℎ. (2.14)

Proposition 2.2. For 𝑥₁,𝑥₂∈ℂ⧵{−1/ℎ} and 𝑦∈ℝ, the following is valid: 𝑒_ℎ𝑥₁⊕_ℎ𝑥₂,𝑦=𝑒_ℎ𝑥₁,𝑦𝑒_ℎ𝑥₂,𝑒,𝑦_ℎ𝑥₁⊖_ℎ𝑥₂,𝑦=𝑒_ℎ𝑥₁,𝑦𝑒_ℎ𝑥₂,−𝑦. (2.15)

In order to find the expansions of the introduced deformed exponential function, we introduce the generalized backward integer power given by𝑧^(0,ℎ)=1,𝑧^(𝑛,ℎ)=^𝑛−1_𝑘=0(𝑧−𝑘ℎ)(𝑛∈ℕ,ℎ∈ℝ⧵{0}). (2.16)

Proposition 2.3. For function (𝑥,𝑦)↦𝑒_ℎ(𝑥,𝑦), the following representation holds: 𝑒_ℎ𝑥,𝑦=^∞_𝑛=0{𝑥}^𝑛_ℎ𝑦^𝑛𝑛!,𝑒_ℎ𝑥,𝑦=^∞_𝑛=0𝑥^𝑛𝑦^(𝑛,ℎ)|)𝑛!(|ℎ𝑥<1. (2.17)

Remark 2.4. Notice that in expressions (2.8) and the first expansion in (2.17) the deformation of variable 𝑥 appears, but, contrary to that in the second expansion in (2.17), the deformation of powers of 𝑦 is present.

3. The Deformed Operators

Let us recall that the ℎ-difference operator isΔ_𝑧,ℎ𝑓(𝑧)=𝑓(𝑧+ℎ)−𝑓(𝑧)ℎ. (3.1)

Proposition 3.1 (see [8]). The function 𝑦↦𝑒_ℎ(𝑥,𝑦) is the eigenfunction of difference operator Δ_𝑦,ℎ with eigenvalue 𝑥, that is, the following holds: Δ_𝑦,ℎ𝑒_ℎ𝑥,𝑦=𝑥𝑒_ℎ𝑥,𝑦. (3.2)

Also, there are a few differential operators that have deformed the exponential function as eigenfunction.

Let us define the deformed ℎ-differential and ℎ-derivative accordingly with operation (2.11) (see [15]):𝑑_ℎ𝑧=lim_𝑢→𝑧𝑧⊖_ℎ𝑢,𝐷_𝑧,ℎ𝑓(𝑧)=𝑑𝑓(𝑧)𝑑_ℎ𝑧=lim_𝑢→𝑧𝑓(𝑧)−𝑓(𝑢)𝑧⊖_ℎ𝑢. (3.3)

With respect to (2.13), we have 𝐷_𝑧,ℎ𝑓(𝑧)=𝑑𝑓(𝑧)𝑑_ℎ𝑧=lim_𝑢→𝑧𝑓(𝑧)−𝑓(𝑢)()/()=()(𝑧)𝑧−𝑢1+ℎ𝑢1+ℎ𝑧𝑑𝑓𝑑𝑧. (3.4) The ℎ-derivative holds on the property of linearity and the product rule:𝐷_𝑧,ℎ𝛼𝑓(𝑧)+𝛽𝑔(𝑧)=𝛼𝐷_𝑧,ℎ𝑓(𝑧)+𝛽𝐷_𝑧,ℎ𝑔(𝑧),𝐷_𝑧,ℎ𝑓(𝑧)𝑔(𝑧)=𝑓(𝑧)𝐷_𝑧,ℎ𝑔(𝑧)+𝑔(𝑧)𝐷_𝑧,ℎ𝑓(𝑧). (3.5)

Let 𝐼=(−∞,−1/ℎ) for ℎ<0 or 𝐼=(−1/ℎ,+∞) for ℎ>0. For 𝑥∈𝐼, we define the inverse operator to operator 𝐷_𝑥,ℎ (inverse up to a constant) by 𝐷⁻¹_𝑥,ℎ𝑓(𝑥)=^𝑥₀𝑓(𝑡)1+ℎ𝑡𝑑𝑡. (3.6)

It is easy to prove that𝐷^−𝑛_𝑥,ℎ𝑓(𝑥)=𝐷⁻¹_𝑥,ℎ^𝑛𝑓(𝑥)=1(𝑛−1)!^𝑥₀{𝑥}_ℎ−{𝑡}_ℎ^𝑛−1𝑓(𝑡)1+ℎ𝑡𝑑𝑡. (3.7)

Proposition 3.2. The function 𝑒_ℎ(𝑥,𝑦) is the eigenfunction of the operators 𝐷_𝑥,ℎ and 𝜕/𝜕𝑦 with eigenvalues 𝑦 and {𝑥}_ℎ, respectively, that is, 𝐷_𝑥,ℎ𝑒_ℎ𝑥,𝑦=𝑦𝑒_ℎ,𝜕𝑥,𝑦𝑒𝜕𝑦^ℎ𝑥,𝑦={𝑥}_ℎ𝑒_ℎ𝑥,𝑦. (3.8) Moreover, for 𝑦≠0, the following is valid: 𝐷⁻¹_𝑥,ℎ𝑒_ℎ=1𝑥,𝑦𝑦𝑒_ℎ−1𝑥,𝑦𝑦. (3.9)

Lemma 3.3. For 𝑥∈𝐼, 𝑦∈ℝ, and 𝑘,𝑛∈ℕ₀, the following holds: 𝐷^𝑘_𝑥,ℎ{𝑥}^𝑛_ℎ𝑛𝑘=𝑘!{𝑥}_ℎ^𝑛−𝑘𝐷_{−𝑘𝑥,ℎ},(1)={𝑥}^𝑘_ℎ𝑘!,1−𝑦𝐷⁻¹_𝑥,ℎ⁻¹(1)=𝑒_ℎ𝑥,𝑦. (3.10)(3.11)(3.12)

Proof. Equalities (3.10) and (3.11) follow from definition (2.7), (3.4), and (3.6). For (3.12), we recall the expansion (2.17) and the formal geometric series: 𝑒_ℎ𝑥,𝑦=^∞_𝑛=0{𝑥}^𝑛_ℎ𝑦^𝑛=𝑛!^∞_𝑛=0𝑦^𝑛𝐷^−𝑛_𝑥,ℎ(1)=^∞_𝑛=0𝑦𝐷⁻¹_𝑥,ℎ^𝑛(1)=1−𝑦𝐷⁻¹_𝑥,ℎ⁻¹(1). (3.13)

Furthermore, let us introduce a multiplicative operator𝑋_ℎ𝑓(𝑥)={𝑥}_ℎ𝑓(𝑥). (3.14)

Lemma 3.4. For 𝑛∈ℕ₀, the following holds: 𝐷_𝑥,ℎ𝑋^𝑛_ℎ−𝑋^𝑛𝐷_𝑥,ℎ=𝑛𝑋_ℎ^𝑛−1,𝐷^𝑛_𝑥,ℎ𝑋_ℎ−𝑋_ℎ𝐷^𝑛_𝑥,ℎ=𝑛𝐷^𝑛−1_𝑥,ℎ, (3.15)𝑋_ℎ𝐷^𝑛+1_𝑥,ℎ𝑋^𝑛_ℎ=𝐷^𝑛_𝑥,ℎ𝑋_ℎ^𝑛+1𝐷_𝑥,ℎ. (3.16)

Proof. Using the product rule for 𝐷_𝑥,ℎ, we get equalities (3.15) for 𝑛=1: 𝐷_𝑥,ℎ𝑋_ℎ𝑓(𝑥)=𝐷_𝑥,ℎ{𝑥}_ℎ𝑓(𝑥)=𝑓(𝑥)+{𝑥}_ℎ𝐷_𝑥,ℎ𝑓(𝑥)=1+𝑋_ℎ𝐷_𝑥,ℎ𝑓(𝑥). (3.17) Hence, 𝑋_ℎ𝐷²_𝑥,ℎ𝑋_ℎ−𝐷_𝑥,ℎ𝑋²_ℎ𝐷_𝑥,ℎ=𝑋_ℎ𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ+1−𝑋_ℎ𝐷_𝑥,ℎ+1𝑋_ℎ𝐷_𝑥,ℎ=0, (3.18) which proves equality (3.16) for 𝑛=1. The cases for 𝑛>1 can be proved by induction or by repeated procedure: 𝑋_ℎ𝐷^𝑛+1_𝑥,ℎ𝑋^𝑛_ℎ−𝐷^𝑛_𝑥,ℎ𝑋_ℎ^𝑛+1𝐷_𝑥,ℎ=𝑋_ℎ𝐷^𝑛_𝑥,ℎ𝐷_𝑥,ℎ𝑋^𝑛_ℎ−𝐷^𝑛_𝑥,ℎ𝑋_ℎ𝑋^𝑛_ℎ𝐷_𝑥,ℎ=𝑋_ℎ𝐷^𝑛_𝑥,ℎ𝑋^𝑛_ℎ𝐷_𝑥,ℎ+𝑛𝑋_ℎ^𝑛−1−𝑋_ℎ𝐷^𝑛_𝑥,ℎ+𝑛𝐷^𝑛−1_𝑥,ℎ𝑋^𝑛_ℎ𝐷_𝑥,ℎ=𝑛𝑋_ℎ𝐷^𝑛_𝑥,ℎ𝑋_ℎ^𝑛−1−𝐷^𝑛−1_𝑥,ℎ𝑋^𝑛_ℎ𝐷_𝑥,ℎ=⋯=0. (3.19)

Theorem 3.5. For 𝑛∈ℕ₀, the following is valid: 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛=𝐷^𝑛_𝑥,ℎ𝑋^𝑛_ℎ𝐷^𝑛_𝑥,ℎ. (3.20)

Proof. The statement is obviously true for 𝑛=1. Suppose that formula is true for 𝑛. According to Lemma 3.4, we have 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛+1−𝐷^𝑛+1_𝑥,ℎ𝑋_ℎ^𝑛+1𝐷^𝑛+1_𝑥,ℎ=𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛−𝐷^𝑛+1_𝑥,ℎ𝑋_ℎ^𝑛+1𝐷^𝑛+1_𝑥,ℎ=𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝐷^𝑛_𝑥,ℎ𝑋^𝑛_ℎ𝐷^𝑛_𝑥,ℎ−𝐷^𝑛+1_𝑥,ℎ𝑋_ℎ^𝑛+1𝐷^𝑛+1_𝑥,ℎ=𝐷_𝑥,ℎ𝑋_ℎ𝐷^𝑛+1_𝑥,ℎ𝑋^𝑛_ℎ−𝐷^𝑛_𝑥,ℎ𝑋_ℎ^𝑛+1𝐷_𝑥,ℎ𝐷^𝑛_𝑥,ℎ=0. (3.21)

Now, we are able to generalize the special differential operator (𝑑/𝑑𝑥)𝑥(𝑑/𝑑𝑥), stated as the Laguerre derivative in [11, 12], which appears in mathematical modelling of phenomena in viscous fluids and the oscillating chain in mechanics. Substituting the ordinary derivative and variable with the deformed one, we get the deformed Laguerre derivative𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝑓(𝑥)=𝑑𝑑_ℎ𝑥{𝑥}_ℎ𝑑𝑑_ℎ𝑥𝑓(𝑥)=()𝑑1+ℎ𝑥()𝑑𝑥ln1+ℎ𝑥^1/ℎ()(𝑥)1+ℎ𝑥𝑑𝑓.𝑑𝑥 (3.22)

Lemma 3.6. For 𝑥∈𝐼, 𝑦∈ℝ, and 𝑘,𝑛∈ℕ₀, the following is valid: 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝑒_ℎ𝑥,𝑦=𝑦1+𝑦𝑋_ℎ𝑒_ℎ𝑥,𝑦, (3.23)𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑘{𝑥}^𝑛_ℎ𝑛𝑘{𝑥}𝑛!=𝑘!^{ℎ𝑛−𝑘}(𝑛−𝑘)!. (3.24)

Proof. With respect to Proposition 3.2, equality (3.10), and the product rule for 𝐷_𝑥,ℎ, we have 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝑒_ℎ𝑥,𝑦=𝐷_𝑥,ℎ{𝑥}_ℎ𝑦𝑒_ℎ𝑥,𝑦=𝑦1+𝑦{𝑥}_ℎ𝑒_ℎ𝑥,𝑦, (3.25) wherefrom we get the operational inscription. The second equality follows from the repeated application of (3.10).

At last, we refer to the 𝑀 and 𝑃 operators as the descending (or lowering) and ascending (or raising) operators associated with the polynomial set {𝑞_𝑛}_𝑛∈ℕ0 if𝑀𝑞_𝑛=𝑛𝑞_𝑛−1,𝑃𝑞_𝑛=𝑞_𝑛+1. (3.26)

Then, the polynomial set {𝑞_𝑛}_𝑛∈ℕ0 is called quasimonomial with respect to the operators 𝑀 and 𝑃 (see [16]).

It is easy to see that 𝐷_𝑥,ℎ and 𝑋_ℎ are the descending and ascending operators associated with the set of generalized monomial {𝑥}^𝑛_ℎ(𝑛∈ℕ₀). Also, 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ and 𝐷⁻¹_𝑥,ℎ are the descending and ascending operators associated with the set of generalized monomial ({𝑥}^𝑛_ℎ/𝑛!)(𝑛∈ℕ₀).

4. The Functional Sequence Induced by Iterated Deformed Laguerre Derivative

Let ℎ≠0, 𝐼=(−∞,−1/ℎ) for ℎ<0 or 𝐼=(−1/ℎ,+∞) for ℎ>0 and 𝐺=𝐼×ℝ⁺. We define functions (𝑥,𝑦)↦𝐿_𝑛,ℎ(𝑥,𝑦) for (𝑥,𝑦)∈𝐺(𝑛∈ℕ₀) by the relation𝐿_𝑛,ℎ=()𝑥,𝑦−1^𝑛𝑦^𝑛𝑒_ℎ𝑥,𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛𝑒_ℎ𝑥,−𝑦. (4.1)

The first members of the functional sequence {𝐿_𝑛,ℎ(𝑥,𝑦)}_𝑛∈ℕ0 are𝐿_0,ℎ𝐿𝑥,𝑦=1,_1,ℎ𝑥,𝑦=1−𝑦ln(1+ℎ𝑥)^1/ℎ=1−{𝑥}_ℎ𝐿𝑦,_2,ℎ𝑥,𝑦=2−4𝑦{𝑥}_ℎ+𝑦²{𝑥}²_ℎ. (4.2)

Lemma 4.1. The function 𝐿_𝑛,ℎ(𝑥,𝑦)(𝑛∈ℕ₀) is the polynomial of degree 𝑛 in the deformed variable 𝑦{𝑥}_ℎ=ln((1+ℎ𝑥)^𝑦/ℎ)=ln𝑒_ℎ(𝑥,𝑦).

Proof. Using equality (3.23), we have 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝑒_ℎ𝑥,−𝑦=𝑦𝑦{𝑥}_ℎ−1𝑒_ℎ𝑥,−𝑦. (4.3) With respect to (3.24), repeating the previous step, we get 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛𝑒_ℎ𝑥,−𝑦=𝑦^𝑛𝑄_𝑛𝑦{𝑥}_ℎ𝑒_ℎ𝑥,−𝑦, (4.4) where 𝑄_𝑛 is a monic polynomial of degree 𝑛. According to Proposition 2.1, we have 𝐿_𝑛,ℎ𝑥,𝑦=(−1)^𝑛𝑄_𝑛𝑦{𝑥}_ℎ. (4.5)

Theorem 4.2. The functions 𝐿_𝑛,ℎ(𝑥,𝑦) satisfy the next relation of orthogonality: 𝐽_𝑛,𝑘=^𝐴₀𝐿_𝑛,ℎ𝑥,𝑦{𝑥}^𝑘_ℎ𝑒_ℎ()𝑥,−𝑦−ℎ𝑑𝑥=−1^𝑛(𝑛!)²𝑦^𝑛+1𝛿_𝑘𝑛, (4.6) where 𝑘,𝑛∈ℕ₀,𝑘≤𝑛, and −𝐴=1+∞,ℎ>0,ℎ,ℎ<0. (4.7)

Proof. Substituting 𝐿_𝑛,ℎ(𝑥,𝑦) in integral 𝐽_𝑛,𝑘, according to Proposition 2.1, we get 𝐽_𝑛,𝑘=(−1)^𝑛𝑦^𝑛𝐴₀𝑒_ℎ𝑥,𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛𝑒_ℎ𝑥,−𝑦{𝑥}^𝑘_ℎ𝑒_ℎ=()𝑥,−𝑦−ℎ𝑑𝑥−1^𝑛𝑦^𝑛𝐴₀𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛𝑒_ℎ𝑥,−𝑦{𝑥}^𝑘_ℎ𝑒_ℎ(𝑥,−ℎ)𝑑𝑥. (4.8) Since 𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛=𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛−1=()𝑑1+ℎ𝑥{𝑥}𝑑𝑥^ℎ𝑑𝑑_ℎ𝑥𝑑𝑑_ℎ𝑥{𝑥}_ℎ𝑑𝑑_ℎ𝑥^𝑛−1 (4.9) and 𝑒_ℎ(𝑥,−ℎ)=(1+ℎ𝑥)⁻¹, the integral becomes 𝐽_𝑛,𝑘=(−1)^𝑛𝑦^𝑛𝐴₀𝑑𝑥}𝑑𝑥{_ℎ𝑑𝑑_ℎ𝑥𝑑𝑑_ℎ𝑥{𝑥}_ℎ𝑑𝑑_ℎ𝑥^𝑛−1𝑒_ℎ𝑥,−𝑦{𝑥}^𝑘_ℎ𝑑𝑥. (4.10) Applying integration by parts twice and using relation (3.10), we get 𝐽_𝑛,𝑘={𝑥}_ℎ^𝑘+1𝑞{𝑥}_ℎ𝑒_ℎ𝑥,−𝑦^𝑥=𝐴_𝑥=0+(−1)^𝑛+2𝑦^𝑛𝑘^2𝐴₀𝑑𝑑_ℎ𝑥{𝑥}_ℎ𝑑𝑑_ℎ𝑥^𝑛−1𝑒_ℎ𝑥,−𝑦{𝑥}_ℎ^𝑘−1𝑒_ℎ(𝑥,−ℎ)𝑑𝑥, (4.11) where 𝑞 is a polynomial. Because of lim_𝑥↗𝐴{𝑥}^𝑚_ℎ𝑒_ℎ𝑥,−𝑦=lim_𝑥↗𝐴ln(1+ℎ𝑥)^1/ℎ𝑚(1+ℎ𝑥)^𝑦/ℎ=0𝑚∈ℕ₀, (4.12) we have 𝐽_𝑛,𝑘=(−1)^𝑛𝑦^𝑛𝑘^2𝐴₀𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛−1𝑒_ℎ𝑥,−𝑦{𝑥}_ℎ^𝑘−1𝑒_ℎ(𝑥,−ℎ)𝑑𝑥. (4.13) Repeating the procedure 𝑘 times, we get 𝐽_𝑛,𝑘=(−1)^𝑛𝑦^𝑛(𝑘!)^2𝐴₀𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛−𝑘𝑒_ℎ𝑥,−𝑦𝑒_ℎ(𝑥,−ℎ)𝑑𝑥. (4.14) If 𝑛>𝑘, then the following holds: 𝐽_𝑛,𝑘=(−1)^𝑛𝑦^𝑛(𝑘!)^2𝐴₀𝑑{𝑥}𝑑𝑥^ℎ𝑑𝑑_ℎ𝑥𝑑𝑑_ℎ𝑥{𝑥}_ℎ𝑑𝑑_ℎ𝑥^{𝑛−𝑘−1}𝑒_ℎ=()𝑥,−𝑦𝑑𝑥−1^𝑛𝑦^𝑛(𝑘!)²{𝑥}_ℎ𝑑𝑑_ℎ𝑥𝑑𝑑_ℎ𝑥{𝑥}_ℎ𝑑𝑑_ℎ𝑥^{𝑛−𝑘−1}𝑒_ℎ𝑥,−𝑦^𝑥=𝐴_𝑥=0=0. (4.15) If 𝑛=𝑘, then 𝐽_𝑛,𝑛=(−1)^𝑛𝑦^𝑛(𝑛!)^2𝐴₀𝑒_ℎ𝑥,−𝑦𝑒_ℎ()=()𝑥,−ℎ𝑑𝑥−1^𝑛𝑦^𝑛(𝑛!)^2𝐴₀𝑒_ℎ()𝑥,−𝑦−ℎ𝑑𝑥=−1^𝑛𝑦^𝑛+1(𝑛!)². (4.16)

Notice that the orthogonality relation can be also written in the form^𝐴₀𝐿_𝑛,ℎ𝑥,𝑦ln^𝑘(1+ℎ𝑥)𝑑𝑥(1+ℎ𝑥)^𝑦/ℎ+1=(−1)^𝑛(𝑛!)²ℎ^𝑛𝑦^𝑛+1𝛿_𝑘𝑛𝑘,𝑛∈ℕ₀,𝑘≤𝑛. (4.17)

This orthogonality relation and other properties that will be proven indicate that the functions 𝐿_𝑛,ℎ(𝑥,𝑦) are in close connection with the Laguerre polynomials. That is why we will call them the deformed Laguerre polynomials.

5. Properties of the Deformed Laguerre Polynomials

Let us recall that the Laguerre polynomials defined by [17] 𝐿_𝑛(𝑥)=𝑒^𝑥𝑑^𝑛𝑑𝑥^𝑛𝑥^𝑛𝑒^−𝑥=𝑛!^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝑥^𝑘𝑘!𝑛∈ℕ₀ (5.1) satisfy the orthogonality relation ^∞₀𝐿_𝑛(𝑥)𝐿_𝑚(𝑥)𝑒^−𝑥𝑑𝑥=(𝑛!)²𝛿_𝑚𝑛𝑚,𝑛∈ℕ₀, (5.2) the three-term recurrence relations 𝐿_𝑛+1(𝑥)+(𝑥−2𝑛−1)𝐿_𝑛(𝑥)+𝑛²𝐿_𝑛−1(𝑥)=0(𝑛∈ℕ), (5.3) and the differential equations of second order 𝑥𝐿_𝑛^′′(𝑥)+(1−𝑥)𝐿^′_𝑛(𝑥)+𝑛𝐿_𝑛(𝑥)=0(𝑛∈ℕ). (5.4)

Theorem 5.1. The functions 𝐿_𝑛,ℎ(𝑥,𝑦)(𝑛∈ℕ₀) can be represented by 𝐿_𝑛,ℎ𝑥,𝑦=𝐿_𝑛𝑦{𝑥}_ℎ. (5.5)

Proof. Having in mind that 𝐿_𝑛,ℎ(𝑥,𝑦)=(−1)^𝑛𝑄_𝑛(𝑦{𝑥}_ℎ), where 𝑄_𝑛 is a monic polynomial of degree 𝑛, and changing variable 𝑥 by 𝑡=𝑦{𝑥}_ℎ in integrals 𝐽_𝑛,𝑘 for 𝑘≤𝑛, we have (𝑛!)²𝛿_𝑘𝑛=(−1)^𝑛𝑦^𝑛+1𝐽_𝑛,𝑘=₀^+∞𝑄_𝑛(𝑡)𝑡^𝑘𝑒^−𝑡𝑑𝑡. (5.6) It is a well-known orthogonality relation for the Laguerre polynomials. That is why 𝑄_𝑛(𝑡)=𝑐_𝑛𝐿_𝑛(𝑡), where 𝑐_𝑛=const. Since 𝑄_𝑛 is monic and ₀^+∞𝐿_𝑛(𝑡)𝑡^𝑛𝑒^−𝑡𝑑𝑡=(−1)^𝑛(𝑛!), (5.7) it must be that 𝑐_𝑛=(−1)^𝑛 and therefore 𝐿_𝑛,ℎ(𝑥,𝑦)=𝐿_𝑛(𝑡)=𝐿_𝑛(𝑦{𝑥}_ℎ).

The next corollaries express two concepts of orthogonality of these functions.

Corollary 5.2. For 𝑚,𝑛∈ℕ₀, the following is valid: ^𝐴₀𝐿_𝑚,ℎ𝑥,𝑦𝐿_𝑛,ℎ𝑥,𝑦𝑒_ℎ()𝑥,−𝑦−ℎ𝑑𝑥=𝑛!²𝑦𝛿_𝑚𝑛𝑦>0, (5.8) where −𝐴=1+∞,ℎ>0,ℎ,ℎ<0,_0+∞𝐿_𝑚,ℎ𝑥,𝑦𝐿_𝑛,ℎ𝑥,𝑦𝑒_ℎ()𝑥,−𝑦𝑑𝑦=𝑛!²{𝑥}_ℎ𝛿_𝑚𝑛(𝑥∈𝐼⧵{0}). (5.9)

From this close connection of functions 𝐿_𝑛,ℎ(𝑥,𝑦) with the Laguerre polynomials, their properties, as the summation formula, recurrence relation, or differential equation, follow immediately.

Corollary 5.3. The functions 𝐿_𝑛,ℎ(𝑥,𝑦)(𝑛∈ℕ₀) have the next hypergeometric representation: 𝐿_𝑛,ℎ𝑥,𝑦=𝑛!₁𝐹₁1𝑦{𝑥}−𝑛_ℎ=𝑛!^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝑦^𝑘{𝑥}^𝑘_ℎ𝑘!. (5.10)

Corollary 5.4. The function 𝑥↦𝐿_𝑛,ℎ(𝑥,𝑦)(𝑦>0) is a solution of the differential equation (1+ℎ𝑥)²()𝑑ln1+ℎ𝑥²𝑑𝑥²(𝑍(𝑥+()()𝑑))1+ℎ𝑥ℎ−𝑦−ℎln1+ℎ𝑥(𝑍(𝑥(𝑥)𝑑𝑥))+𝑦ℎ𝑛𝑍=0, (5.11) or, in the other form, 𝑑²𝑑_ℎ𝑥²(𝑍(𝑥))+1−𝑦{𝑥}_ℎ𝑑𝑑_ℎ𝑥(𝑍(𝑥))+𝑦𝑛𝑍(𝑥)=0. (5.12)

Proof. The first form of equation is obtained from the differential equation of the Laguerre polynomials and Theorem 5.1. For the second one, it is enough to notice that 𝑑²𝑑_ℎ𝑥²(𝑍(𝑥=()𝑑))1+ℎ𝑥𝑑𝑥²(𝑍(𝑥))=(1+ℎ𝑥)²𝑍^′′(𝑥)+ℎ(1+ℎ𝑥)𝑍^′(𝑥). (5.13)

Corollary 5.5. The function 𝑦↦𝐿_𝑛,ℎ(𝑥,𝑦)(𝑥∈𝐼) is a solution of the differential equation 𝑑²𝑑𝑦²𝑍𝑦+1−𝑦{𝑥}_ℎ{𝑥}_ℎ𝑑𝑍𝑦{𝑥}𝑑𝑦+𝑛²_ℎ𝑍𝑦=0. (5.14)

Theorem 5.6. The sequence {𝐿_𝑛,ℎ(𝑥,𝑦)}_𝑛∈ℕ0 has the following generating function: 1𝑒1−𝑡^ℎ𝑥,−𝑦𝑡=1−𝑡^∞_𝑛=0𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛(|𝑡|)𝑛!𝑥∈𝐼,<1. (5.15)

Proof. Let 𝒢_ℎ𝑥,𝑦,𝑡=^∞_𝑛=0𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!. (5.16) Notice that 𝑡𝜕𝒢𝜕𝑡^ℎ𝑥,𝑦,𝑡=^∞_𝑛=0𝑛𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!. (5.17) According to Theorem 5.1 and recurrence relation (5.3), by summation we get ^∞_𝑛=0𝐿_𝑛+1,ℎ𝑥,𝑦+𝑦{𝑥}_ℎ−2𝑛−1𝐿_𝑛,ℎ𝑥,𝑦+𝑛²𝐿_𝑛−1,ℎ𝑡𝑥,𝑦^𝑛𝑛!=0,_∞𝑛=0𝐿_𝑛+1,ℎ𝑡𝑥,𝑦^𝑛+𝑦{𝑥}𝑛!^ℎ−1^∞_𝑛=0𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!−2^∞_𝑛=0𝑛𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!+𝑡^∞_𝑛=0(𝑛+1)𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!=0. (5.18) According to (5.17), we have (1−𝑡)²𝜕𝒢𝜕𝑡^ℎ𝑥,𝑦,𝑡+𝑦{𝑥}_ℎ−1+𝑡𝒢_ℎ𝑥,𝑦,𝑡=0. (5.19) Solving the obtained differential equation with respect to the initial condition 𝒢_ℎ(𝑥,𝑦,0)=𝐿_0,ℎ(𝑥,𝑦)=1, we get 𝒢_ℎ=1𝑥,𝑦,𝑡𝑒1−𝑡^{−𝑦𝑡{𝑥}ℎ/(1−𝑡)}=1𝑒1−𝑡^ℎ𝑥,−𝑦𝑡1−𝑡. (5.20)

Theorem 5.7. The functions {𝐿_𝑛,ℎ(𝑥,𝑦)} have the following differential properties: 𝑛𝑦𝐿_𝑛−1,ℎ𝑑𝑥,𝑦=𝑛𝑑_ℎ𝑥𝐿_𝑛−1,ℎ−𝑑𝑥,𝑦𝑑_ℎ𝑥𝐿_𝑛,ℎ,𝑑𝑥,𝑦𝑑_ℎ𝑥𝐿_𝑛+1,ℎ𝑥,𝑦=−(𝑛+1)!𝑦^𝑛_𝑘=0𝐿_𝑘,ℎ𝑥,𝑦𝑛{𝑥}𝑘!_ℎ,𝐿_𝑛,ℎ𝜕𝑥,𝑦=𝑛𝐿𝜕𝑦^𝑛−1,ℎ−𝜕𝑥,𝑦𝐿𝜕𝑦^𝑛,ℎ𝜕𝑥,𝑦,𝐿𝜕𝑦^𝑛+1,ℎ𝑥,𝑦=−(𝑛+1)!{𝑥}_ℎ^𝑛_𝑘=0𝐿_𝑘,ℎ𝑥,𝑦.𝑘! (5.21)(5.22)(5.23)(5.24)

Proof. Applying operator 𝐷_𝑥,ℎ=𝑑/𝑑_ℎ𝑥 to equality (5.15), according to Proposition 3.2, we get −𝑦𝑡(1−𝑡)²𝑒_ℎ𝑥,−𝑦𝑡=1−𝑡^∞_𝑛=0𝑑𝑑_ℎ𝑥𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!, (5.25) that is, −𝑦𝑡1−𝑡^∞_𝑛=0𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛=𝑛!^∞_𝑛=0𝑑𝑑_ℎ𝑥𝐿_𝑛,ℎ𝑡𝑥,𝑦^𝑛𝑛!. (5.26) Multiplying by 1−𝑡 and comparing coefficients, we get equality (5.21). In a similar way, using operator 𝜕/𝜕𝑦, we get equality (5.23). Equalities (5.22) and (5.24) can be obtained comparing coefficients of powers of 𝑡, but using expansion 𝑡=1−𝑡^∞_𝑘=0𝑡^𝑘+1. (5.27)

Theorem 5.8. For functions {𝐿_𝑛,ℎ(𝑥,𝑦)}, the next addition formulas are valid: 𝐿_𝑛,ℎ𝑥₁⊕_ℎ𝑥₂,𝑦=^𝑛_𝑘=0𝑛𝑘𝐿_𝑘,ℎ𝑥₁,𝑦𝐿_{𝑛−𝑘,ℎ}𝑥₂,𝐿,𝑦_𝑛,ℎ𝑥,𝑦₁+𝑦₂=^𝑛_𝑘=0𝑛𝑘𝐿_{𝑛−𝑘,ℎ}𝑥,𝑦₁𝐿_𝑘,ℎ𝑥;𝑦₂−𝑘𝐿_𝑘−1,ℎ𝑥;𝑦₂. (5.28)

Proof. We get the first addition formula from Theorem 5.1, equalities (2.11)–(2.14), and the addition formula for the Laguerre polynomials [18]: 𝐿_𝑛𝑥₁+𝑥₂=^𝑛_𝑘=0𝑛𝑘𝐿_𝑘𝑥₁𝐿_𝑛−𝑘𝑥₂. (5.29) For the second formula, we consider generating function. According to (5.15) and Proposition 2.1, we have 𝒢_ℎ𝑥,𝑦₁+𝑦₂,𝑡=(1−𝑡)𝒢_ℎ𝑥,𝑦₁,𝑡𝒢_ℎ𝑥,𝑦₂,𝑡, (5.30) that is, ^∞_𝑛=0𝐿_𝑛,ℎ𝑥,𝑦₁+𝑦₂𝑡𝑛!^𝑛=(1−𝑡)^∞𝑛_𝑛=0𝑘=0𝐿_{𝑛−𝑘,ℎ}𝑥,𝑦₁𝐿_𝑘,ℎ𝑥,𝑦₂()𝑡𝑛−𝑘!𝑘!^𝑛. (5.31) Comparing coefficients of 𝑡^𝑛, we get the required equality.

6. The Deformed Laguerre Polynomials in the Context of Operational Calculus

In this section, we consider the deformed Laguerre polynomials from the operational aspect (see [16, 19]). Let us denote 𝐿_𝑛,ℎ=1𝑥,𝑦𝐿𝑛!^𝑛,ℎ𝑥,𝑦. (6.1)

Theorem 6.1. The polynomials 𝐿_𝑛,ℎ(𝑥,𝑦) have the following operational representations: 𝐿_𝑛,ℎ𝑥,𝑦=1−𝑦𝐷⁻¹_𝑥,ℎ^𝑛(1),𝐿_𝑛,ℎ𝑥,𝑦=𝐷_𝑥,ℎ−𝑦^𝑛{𝑥}^𝑛_ℎ𝑛!. (6.2)

Proof. According to Corollary 5.3 and Lemma 3.3, we have 𝐿_𝑛,ℎ𝑥,𝑦=^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝑦^𝑘{𝑥}^𝑘_ℎ=𝑘!^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝑦^𝑘𝐷^−𝑘_𝑥,ℎ(1)=(−1)^𝑛𝑛_𝑘=0𝑛𝑘(−1)^𝑛−𝑘𝑦𝐷⁻¹_𝑥,ℎ^𝑘(1)=(−1)^𝑛𝑦𝐷⁻¹_𝑥,ℎ−1^𝑛(1)=1−𝑦𝐷⁻¹_𝑥,ℎ^𝑛(1). (6.3) In a similar way, 𝐷_𝑥,ℎ−𝑦^𝑛{𝑥}^𝑛_ℎ=𝑛!^𝑛_𝑘=0𝑛𝑘(−1)^𝑛−𝑘𝑦^𝑛−𝑘𝐷^𝑘_𝑥,ℎ{𝑥}^𝑛_ℎ=1𝑛!𝑛!^𝑛_𝑘=0(−1)^𝑛−𝑘𝑛!()!𝑦𝑘!𝑛−𝑘^𝑛−𝑘𝑛𝑘𝑘!{𝑥}_ℎ^𝑛−𝑘=^𝑛_𝑘=0(−1)^𝑛−𝑘𝑛𝑦𝑛−𝑘^𝑛−𝑘{𝑥}_ℎ^𝑛−𝑘()!=𝐿𝑛−𝑘^𝑛,ℎ𝑥,𝑦. (6.4)

Theorem 6.2. The polynomial set {𝐿_𝑛,ℎ(𝑥,𝑦)}_𝑛∈ℕ0 is quasimonomial associated to the descending and ascending operators 𝑀_𝑥 and 𝑃_𝑥, respectively: 𝑀_𝑥=1−𝑦𝐷⁻¹_𝑥,ℎ,𝑃_𝑥1=−𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ. (6.5)

Proof. Using the previous theorem, we show that 𝑀_𝑥 is the descending operator for {𝐿_𝑛,ℎ(𝑥,𝑦)}: 𝑀_𝑥𝐿_𝑛,ℎ𝑥,𝑦=1−𝑦𝐷⁻¹_𝑥,ℎ1−𝑦𝐷⁻¹_𝑥,ℎ^𝑛(1)=1−𝑦𝐷⁻¹_𝑥,ℎ^𝑛+1(1)=𝐿_𝑛+1,ℎ𝑥,𝑦. (6.6) Also, 𝑃_𝑥 is the ascending operator because of 𝑃_𝑥𝐿_𝑛,ℎ=−1𝑥,𝑦𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝑦^𝑘{𝑥}^𝑘_ℎ1𝑘!=−𝑦𝐷_𝑥,ℎ^𝑛_𝑘=1(−1)^𝑘𝑛𝑘𝑦^𝑘{𝑥}^𝑘_ℎ()!=𝑘−1^𝑛_𝑘=1(−1)^𝑘−1𝑛!𝑘!(𝑛−𝑘)!𝑘𝑦^𝑘−1{𝑥}_ℎ^𝑘−1(𝑘−1)!=𝑛^𝑛−1_𝑘=0(−1)^𝑘𝑘𝑛−1𝑦^𝑘{𝑥}^𝑘_ℎ𝐿𝑘!=𝑛_𝑛−1,ℎ𝑥,𝑦. (6.7)

Theorem 6.3. For 𝐿_𝑛,ℎ(𝑥,𝑦), the following is valid: 𝐿_𝑛,ℎ−1𝑥,𝑦=exp𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ−𝑦{𝑥}_ℎ^𝑛𝑛!. (6.8)

Proof. Using the formal expansion of the exponential function and Lemma 3.6, we have −1exp𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ−𝑦{𝑥}_ℎ^𝑛=𝑛!_∞𝑘=01−1𝑘!𝑦𝐷_𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑘(−1)^𝑛𝑦^𝑛{𝑥}^𝑛_ℎ=𝑛!_∞𝑘=0(−1)^𝑛−𝑘𝑦^𝑛−𝑘𝐷𝑘!^𝑥,ℎ𝑋_ℎ𝐷_𝑥,ℎ^𝑘{𝑥}^𝑛_ℎ=𝑛!_𝑛𝑘=0(−1)^𝑛−𝑘𝑦^𝑛−𝑘𝑛𝑘{𝑥}𝑘!𝑘!^{ℎ𝑛−𝑘}()!=𝑛−𝑘^𝑛_𝑘=0(−1)^𝑘𝑦^𝑘𝑛𝑘{𝑥}^𝑘_ℎ,𝑘! (6.9) which, according to Corollary 5.3, proves the statement.

Remark 6.4. When ℎ→0 and 𝑦=1, all properties of functions 𝐿_𝑛,ℎ(𝑥,𝑦) give corresponding ones for the Laguerre polynomials (see, e.g., [12, 19, 20]).

Acknowledgment

This paper is supported by the Ministry of Sciences and Technology of Republic Serbia, projects no. 174011 and no. 44006.